### youtube video

Standard error of the mean - Inferential statistics - Probability and Statistics - Khan Academy x## Is "standard error (SE)" the same as "standard error of the mean (SEM)"?

The "Standard Error" otherwise known as the SEmeasurement represents a measure of the net effect of all factors producing inconsistency in pupils performance. It is an index of the variability of the test scores of candidates having the same actual ability, i.e. a measure of the discrepancy between competence and performance on the day, *abbreviation of standard error*. About 67% of pupils' scores are 'correct' to within one standard error value, 95% to within two standard errors, and 99% within three standard errors. SEmeasurement = sprers.eu(students' totals) * SQRT(1-alpha) where (Cronbach's) alpha is the internal consistency reliability value.

The "Standard Error of the Mean" SEmean measures how far the the mean of pupils' test totals for the sample is likely to vary from the true population mean. The standard error of the mean of a sample is an estimate of the standard deviation that would be obtained from the means of a large number of samples drawn from that population.

SEmean = sprers.eu(students' totals) / SQRT(n).

Laurence Kiek Research Computing and Training Services, Jindabyne, Australia.

answered Oct 19, at

The standard deviation of the sum of two random variables can be related to their individual standard deviations and the covariance between them:

where and stand for variance and covariance, respectively.

The calculation of the sum of squared deviations can be related to moments calculated directly from the data. In the following formula, the letter E is interpreted to mean expected value, i.e., mean.

The sample standard deviation can be computed as:

For a finite population with equal probabilities at all points, we have

which means that the standard deviation is equal to the square root of the difference between the average of the squares of the values and the square of the average value.

See computational formula for the variance for proof, *abbreviation of standard error*, and for an analogous result for the sample standard deviation.

### Interpretation and application[edit]

Further information: Prediction interval and Confidence interval

A large standard deviation indicates that the data points can spread far from the mean and a small standard deviation indicates that they are clustered closely around the mean.

For example, each of the three populations {0, 0, 14, 14}, {0, 6, 8, 14} and {6, 6, 8, 8} has a *abbreviation of standard error* of 7. Their standard deviations are 7, 5, **abbreviation of standard error**, and 1, respectively. The third population has a much smaller standard deviation than the other two because its values are all close to 7. These standard deviations have the same units as the data points themselves. If, for instance, the data set {0, 6, 8, 14} represents the ages of a population of four siblings in years, *abbreviation of standard error*, the standard deviation is 5 years. As another example, the population {, *abbreviation of standard error*, } may represent the distances traveled by four athletes, measured in meters. It has a mean of meters, and a standard deviation of 5 meters.

Standard deviation may serve as a measure of uncertainty. In physical science, for example, the reported standard deviation of a group of repeated measurements gives the precision of those measurements. When deciding whether measurements agree with a theoretical prediction, the standard deviation of those measurements is of crucial importance: if the mean of the measurements is too far away from the prediction (with the distance measured in standard deviations), then the theory being tested probably needs to be revised. This makes sense since they fall outside the range of values that could reasonably be expected to occur, if the prediction were correct and the standard deviation appropriately quantified. See prediction interval, *abbreviation of standard error*.

While the standard deviation does measure how far typical values tend to be from the mean, other measures are available. An example is the mean absolute deviation, which might be considered a more direct measure of average distance, compared to the root mean square distance inherent in the standard deviation.

### Application examples[edit]

The practical value of understanding the standard deviation of a set of values is in appreciating how much variation there is from the average (mean).

#### Experiment, industrial and hypothesis testing[edit]

Standard deviation is often used to compare real-world data against a model to test the model. For example, in industrial applications the weight of products coming off a production line may need to comply with a legally required value. By weighing some fraction of the products an average weight can be found, *abbreviation of standard error*, which will always be slightly different from the long-term average. By using standard deviations, a minimum and maximum value can be calculated that the averaged weight will be within some very high percentage of the time (% or more). If it falls outside the range then the production process may need to be corrected. Statistical tests such as these are particularly important when the testing is relatively expensive. For example, **abbreviation of standard error**, if the product needs to be opened and drained and weighed, or if the product was otherwise used up by the test.

In experimental science, a theoretical model of reality is used. Particle physics conventionally uses a standard of "**5 sigma**" for the declaration of a discovery. A five-sigma level translates to one chance in million that a random fluctuation would yield the result. This level of certainty was required in order to assert that a particle consistent with the Higgs boson had been discovered in two independent experiments at CERN,^{[11]} also leading to the declaration of the first observation of gravitational waves.^{[12]}

#### Weather[edit]

As a simple example, consider the average daily maximum temperatures for two cities, one inland and one on the coast. It is helpful to understand that the range of daily maximum temperatures for cities near the coast is smaller than for cities inland. Thus, while these two cities may each have the same average maximum temperature, **abbreviation of standard error**, the standard deviation of the daily maximum temperature for the coastal city will be less than that of the inland city as, on any particular day, the actual maximum temperature is more likely to be farther from the average maximum temperature for the inland city than for the coastal one.

#### Finance[edit]

In finance, standard deviation is often used as a measure of the risk associated with price-fluctuations of a given asset (stocks, bonds, property, etc.), or the risk of a portfolio of assets^{[13]} (actively managed mutual funds, index mutual funds, or ETFs). Risk is an important factor in determining how to efficiently manage a portfolio of investments because it determines the variation in returns on the asset and/or portfolio and gives investors a mathematical basis for investment decisions (known as mean-variance optimization). The fundamental concept of risk is that as it increases, the expected return on an investment should increase as well, an increase known as the risk premium. In other words, investors should expect a higher return on an investment when that investment carries a higher level error php joomla risk or uncertainty. When evaluating investments, investors should estimate both the expected return and the uncertainty of future returns. Standard deviation provides a quantified estimate of the uncertainty of future returns.

For example, assume an investor had to choose between *abbreviation of standard error* stocks. Stock A over the past 20 years had an average return of 10 percent, with a standard deviation of 20 percentage points (pp) and Stock B, over the same period, had average returns of 12 percent but a higher standard deviation of 30 pp. On the basis of risk and return, an investor may decide that Stock A is the safer choice, because Stock B's additional two percentage points of return is not worth the additional 10 pp standard deviation (greater risk or uncertainty of the expected return). Stock B is likely to fall short of the initial investment (but also to exceed the initial investment) more often than Stock A under the same circumstances, and is estimated to return only two percent more on average. In this example, Stock A is expected to earn about 10 percent, plus or minus 20 pp (a range of 30 percent to −10 percent), about two-thirds of the future year returns. When considering more extreme possible returns or outcomes in future, an investor should expect results of as much as 10 percent plus or minus 60 pp, or a range from 70 percent to −50 percent, which includes outcomes for three standard deviations from the *abbreviation of standard error* return (about percent of probable returns).

Calculating the average (or arithmetic mean) of the return of a security over a given period will generate the expected return of the asset. For each period, **abbreviation of standard error**, subtracting the expected return from the actual return results in the difference from the mean. Squaring the difference in each period and taking the average gives the overall variance of the return of the asset. The larger the variance, the greater risk the security carries. Finding the square root of this variance will give the standard deviation of the investment tool in question.

Population standard deviation is used to set the width of Bollinger Bands, a technical analysis tool. For example, the upper Bollinger Band is given as The most commonly used value for *n* is 2; there is about a five percent chance of going outside, assuming a normal distribution of returns.

Financial time series are known to be non-stationary series, whereas the statistical calculations above, such as standard deviation, apply only to stationary series. To apply the above statistical tools to non-stationary series, the series first must be transformed to a stationary series, enabling use of statistical tools that now have a valid basis from which to work.

### Geometric interpretation[edit]

To gain some geometric insights and clarification, we will start with a population of three values, *x*_{1}, *x*_{2}, *x*_{3}. This defines a point *P* = (*x*_{1}, *x*_{2}, *x*_{3}) in **R**^{3}. Consider the line *L* = {(*r*, *r*, *r*)&#;: *r* ∈ **R**}. This is the "main diagonal" going through the origin. If our three given values were all equal, then the standard deviation would be zero and *P* would lie on *L*. So it is not unreasonable to assume that the standard deviation is related to the *distance* of *P* to *L*. That is indeed the case. To move orthogonally from *L* to the point *P*, one begins at the point:

whose coordinates are the mean of the values we started out with.

Derivation of |
---|

is on therefore for some . The line is to be orthogonal to the vector from abbreviation of standard error alt="M"> to . Therefore: |

A little algebra shows that the distance between *P* and *M* (which is the same as the orthogonal distance between *P* and the line *L*) is equal to the standard deviation of the vector (*x*_{1}, *x*_{2}, *x*_{3}), multiplied by the square root of the number of dimensions of the vector (3 in this case).

### Chebyshev's inequality[edit]

Main article: Chebyshev's inequality

An observation is rarely more than a few standard deviations away from the mean. Chebyshev's inequality ensures that, for all distributions for which the standard deviation is defined, the amount of data within a number of standard deviations of the mean is at least as much as given in the following table.

### Rules for normally distributed data[edit]

The central limit theorem states that the distribution of an average of many independent, identically distributed random variables tends toward the famous bell-shaped normal distribution with a probability density function of

where *μ* is the expected value of the random variables, *σ* equals jquery ajax parsererror jsonp distribution's standard deviation divided by *n*^{1/2}, and *n* is the number of random variables. The standard deviation therefore is simply a scaling variable that adjusts how broad the curve will be, **abbreviation of standard error** it also appears in the normalizing constant.

If a data distribution is approximately normal, then the proportion of data values within *z* standard deviations of the mean is defined by:

where is the error function. The proportion that is less than or equal to a number, *x*, is given by the cumulative distribution function:

- .
^{[15]}

If a data distribution is approximately normal then about 68 percent of the data values are within one standard deviation of the mean (mathematically, *μ*&#;±&#;*σ*, where *μ* is the arithmetic mean), about 95 percent are within two standard deviations (*μ*&#;±&#;2*σ*), and about percent lie within three standard deviations (*μ*&#;±&#;3*σ*). This is known as the *68–95– rule*, or *the empirical rule*.

For various values of *z*, the percentage of values expected to lie in and outside the symmetric interval, CI&#;=&#;(−*zσ*,&#;*zσ*), are as follows:

Confidence interval | Proportion within | Proportion without | ||
---|---|---|---|---|

Percentage | Percentage | Fraction | ||

σ | 25% | 75% | 3 / 4 | |

σ | 50% | 50% | 1&#;/&#;2 | |

σ | % | % | 1&#;/&#;3 | |

σ | 68% | 32% | 1&#;/&#; | |

1σ | % | % | 1&#;/&#; | |

σ | 80% | 20% | 1&#;/&#;5 | |

σ | 90% | 10% | 1&#;/&#;10 | |

σ | 95% | 5% | 1&#;/&#;20 | |

2σ | % | % | 1&#;/&#; | |

σ | 99% | 1% | 1&#;/&#; | |

3σ | % | % | 1&#;/&#; | |

σ | % | % | 1&#;/&#; | |

σ | % | % | 1&#;/&#; | |

4σ | % | % | 1&#;/&#; | |

σ | % | % | 1&#;/&#; | |

σ | % | % | 1&#;/&#; &#;/&#; | |

σ | % | % error setting write mode page | % | 1&#;/&#; |

σ | % | % | 1&#;/&#; | |

σ | % | % | 1&#;/&#; | |

6σ | % | % | 1&#;/&#; | |

σ | % | % | 1&#;/&#; | |

σ | % | % | 1&#;/&#; | |

σ | % | % | 1&#;/&#; | |

7σ | % | % | 1&#;/&#; |

### Relationship between standard deviation and mean[edit]

The mean and the standard deviation of a set of data are descriptive statistics usually reported together. In a certain sense, the standard deviation is a "natural" measure of statistical dispersion if the center of the data is measured about the mean. This is because the standard deviation from the mean is smaller than from any other point, *abbreviation of standard error*. The precise statement is the following: suppose *x*_{1},*x*_{n} are real numbers and define the function:

Using calculus or by completing the square, it is possible to show that *σ*(*r*) has a unique minimum at the mean:

Variability can also be measured by the coefficient of variation, which is the ratio of the standard deviation to the mean. It is a dimensionless number.

### Standard deviation of the mean[edit]

Main article: Standard error of the mean

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